In realistic ocean flows, time-dependent coherent structures, or Lagrangian coherent structures (LCS), are similar to separatrices that divide the flow into dynamically distinct regions. LCS are extensions of stable and unstable manifolds to general time-dependent flows (see, e.g., G. Haller and G. Yuan, “Lagrangian coherent structures and mixing in two-dimensional turbulence,” Phys. D, vol. 147, pp. 352-370 (December 2000)) and they carry a great deal of global information about the dynamics of the flows.
For two-dimensional (2D) flows, LCS are analogous to ridges defined by local maximum instability, and can be quantified by local measures of Finite-Time Lyapunov Exponents (FTLE) (S. C. Shadden, F. Lekien, and J. Marsden, “Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,” Physica D: Nonlinear Phenomena, vol. 212, no. 3-4, pp. 271-304 (2005)). Recently, LCS have been shown to coincide with optimal trajectories in the ocean which minimize the energy and the time needed to traverse from one point to another (see, e.g., T. Inane, S. Shadden, and J. Marsden, “Optimal trajectory generation in ocean flows,” in Proceedings of the 2005 American Control Conference, pp. 674-679, 2005, and C. Senatore and S. Ross, “Fuel-efficient navigation in complex flows,” in Proceedings of the 2008 American Control Conference, pp. 1244-1248, 2008). Furthermore, to improve weather and climate forecasting, and to better understand various physical, chemical, and geophysical processes in the ocean, there has been significant interest in the deployment of autonomous sensors to measure a variety of quantities of interest. One drawback to operating sensors in time-dependent and stochastic environments like the ocean is that the sensors will tend to escape from their monitoring region of interest. Since the LCS are inherently unstable and denote regions of the flow where more escape events may occur (see, e.g., E. Forgoston, L. Billings, P. Yecko, and I. B. Schwartz, “Set-based corral control in stochastic dynamical systems: Making almost invariant sets more invariant,” Chaos, vol. 21, 013116, 2011), knowledge of the LCS are of paramount importance in maintaining a sensor in a particular monitoring region.
Existing work in cooperative boundary tracking for robotic teams that relies on one-dimensional (1D) parameterizations include C. Hsieh, Z. Jin, D. Marthaler, B. Nguyen, D. Tung, A. Bertozzi, and R. Murray, “Experimental validation of an algorithm for cooperative boundary tracking,” in Proceedings of the 2005 American Control Conference, pp. 1078-1083,2005, S. Susca, S. Martinez, and F. Bullo, “Monitoring environmental boundaries with a robotic sensor network,” IEEE Trans. on Control Systems Technology, vol. 16, no. 2, pp. 288-296, 2008, I. Triandaf and I. B. Schwartz, “A collective motion algorithm for tracking time-dependent boundaries,” Mathematics and Computers in Simulation, vol. 70, pp. 187-202, (2005) and V. M. Goncalves, L. C. A. Pimenta, C. A. Maia, B. Dutra, and G. A. S. Pereira, “Vector fields for robot navigation along time-varying curves in n-dimensions,” IEEE Trans. on Robotics, vol. 26, no. 4, pp. 647-659 (2010), for static and time-dependent cases respectively. Formation control strategies for distributed estimation of level surfaces and scalar fields in the ocean are presented in F. Zhang, D. M. Fratantoni, D. Paley, J. Lund, and N. E. Leonard, “Control of coordinated patterns for ocean sampling,” Int. Journal of Control, vol. 80, no. 7, pp. 1186-1199 (2007), K. M. Lynch, P. Schwartz, I. B. Yang, and R. A. Freeman, “Decentralized environmental modeling by mobile sensor networks,” IEEE Trans. on Robotics, vol. 24, no. 3, pp. 710-724 (2008), and W. Wu and F. Zhang, “Cooperative exploration of level surfaces of three dimensional scalar fields,” Automatica, the IFAC Journal, vol. 47, no. 9, pp. 2044-2051 (2011), and pattern formation for surveillance and monitoring by robot teams is discussed in J. Spletzer and R. Fierro, “Optimal positioning strategies for shape changes in robot teams,” in Proceedings of the IEEE Int. Conf. on Robotics & Automation, Barcelona, Spain pp. 754-759, 2005, S. Kalantar and U. R. Zimmer, “Distributed shape control of homogeneous swarms of autonomous underwater vehicles,” Autonomous Robots (intl. journal), 2006, and M. A. Hsieh, S. Loizou, and V. Kumar, “Stabilization of multiple robots on stable orbits via local sensing,” in Proceedings of the Int. Conf. on Robotics & Automation (ICRA), 2007).